Download - SN- Lecture 11
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Contagion On Social Networks
Lecture 11
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To understand:
Aims Lecture 11
The functioning of contagion in networks
Some properties of real life social networks
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How does network structure impact behavior?
Networks & Behavior
In this lecture we will take the networks as given
We are going to see the effect of a network on behavior
Simple infections, contagion, diffusion (1 or 0)
Choices, decisions - games on networks (strategic interaction )
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Next Lecture
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However
Keep in mind
The relation between the outcomes we can get for a given network will affect which links we form
There is a co-determination between structure & behavior
The macro-micro-macro link
For this course, we will look at them separately
At the end of the course we will say something more about it
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Example
Contagion
In a girls dormitory in college, he asked students about their friendship: specifically in the dinning table
Jacob Moreno (1960)
Two choices (number 1 and number 2 friends you dine with)Who do you dine with?
He put all the data into a network
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Example
Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
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Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
Ada is my first choice and Jean my
second
Direction in the connections
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Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
Ada is my first choice and Jean my
second
Hellen & Robin are my choices
Direction in the connections
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Who is the most popular girl?
Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
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Who is the most popular girl?
Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
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In directed networks
Variants of degree
In-degree
Number of links from others to me
Out-degree
Number of links from me to others
Reciprocity
I choose a person who also chooses me
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Isolated Group
Contagion
1
1
Cora
2
Jean
Hellen
Robin2
11
Ada
2
1
2 2
Louise Lena
Marion
21
2
Eva
2
1
Martha
21
1
2
1
2
2
Adele
Maxine
Frances1
12
Anna1
Alice Laura
Ella
2
1
Ellen2 1
Edna
1 21 2
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
2
11
22
1
21
2
12
1
21
1
2
No reciprocityout degree>in degree
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Size of these isolated groups
Important
If you are in an isolated group, and a disease outbreaks, it might get
stucked in those isolated locations
Measures of connectivity & navigation in the network are fundamental for problems of
contagion
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Always mutual
Contact
Cora
Jean
Hellen
Robin
Ada
Louise Lena
Marion
Eva
Martha
Adele
Maxine
Anna
Alice Laura
Ella
Ellen
Edna
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
Frances
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Who are the people in more danger?Contagion
Cora
Jean
Hellen
Robin
Ada
Louise Lena
Marion
Eva
Martha
Adele
Anna
Alice Laura
Ella
Ellen
Edna
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
Frances
Maxine
I am sick
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& after that?Contagion
Cora
Jean
Hellen
Robin
Ada
Louise Lena
Marion
Eva
Martha
Adele
Anna
Alice Laura
Ella
Ellen
Edna
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
Frances
Maxine
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& after that?Contagion
Cora
Jean
Hellen
Robin
Ada
Louise Lena
Marion
Eva
Martha
Adele
Anna
Alice Laura
Ella
Ellen
Edna
Mary
Jane
Hazel
Betty
Hilda
Ruth
Irene
Frances
Maxine
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Contagion
The previous were just some questions you can answer using networks
More to come up, but first an example
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Transmission NetworkExample
https://www.youtube.com/watch?v=VZGHGVIedzA
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What’s the extent of diffusion?
Other Questions
How does it depend on the process as well as the network?
Is everyone infected?
Are some network architechtures more suitable for contagion?
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Low density - no contagion
Some main results
Part of the population infected
Degree affects who is infected & when
Middles density - some probability of contagion
High density - sure infection & all infected
Network structure matters
This is only one side of the problem
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How do they look like?Real life networks
It is important to know what kind of networks allow transmission to flow better/worse
But, how do real life networks relate to this?Are there universal structural properties?
Every network is unique microscopically, but with appropriate definitions, stricking macroscopic commonalities emerge
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Large scale networksProperties
Main claim:Typical large scale networks exhibit:
Heavy-tailed degree distributions
Small diameter
High clustering
Hubs or connectors
Six degrees of separation?
Friends of friends are friends
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degree distributionsHeavy-tailed
Lots of nodes with small degree and few nodes with very high degree
Degree
Number of nodes
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degree distributionsHeavy-tailed
Erdös Number Projecthttp://www.oakland.edu/enp/
Paul Erdös1913-1996
Collaboration network between mathematicians
Nodes are mathematiciansLink if they coautor a research paper togetherPaul Erdös is in the centerThe number of a node is her distance to P.E.P.E. has an Erdös-number = 0A coauthor of P.E. has Erdös-number = 1
Their coauthors = 2, and so on...
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Network of P.E.’s coauthorsErdös-number
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Degree distributions
Details:
Size (N)
410,000 authorsSize (g)
676,000 links
Average (di)
3.36
Erdös number 0 --- 1 personErdös number 1 --- 504 peopleErdös number 2 --- 6593 peopleErdös number 3 --- 33605 peopleErdös number 4 --- 83642 peopleErdös number 5 --- 87760 peopleErdös number 6 --- 40014 peopleErdös number 7 --- 11591 peopleErdös number 8 --- 3146 peopleErdös number 9 --- 819 peopleErdös number 10 --- 244 peopleErdös number 11 --- 68 peopleErdös number 12 --- 23 peopleErdös number 13 --- 5 people
Diameter (N,g)
7.64
Similar network for acting & Kevin Bacon
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Compared to the population sizeSmall Diameter
Arguably, every person in the world is at diameter 6 from anyone else
http://www.youtube.com/watch?v=HLIyuYwbVnA
Think about Milgram’s experiment with the letters
Other Networks:Messenger (Lescovec & Horvitz, 2008)
Diameter = 6.5; N = 180 millionsFacebook (Backstrom et al., 2012)
Diameter = 5; N = 721 millions
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Compared to average degreeHigh Clustering
How likely two nodes that share a common neighbor are to be neighbors themselves
Examples Networks:(Watts, 2003)
Movie actor networkC.C.=0.79 ; p=0.00027Neuronal networkC.C.=0.28 ; p=0.05
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Checklist
Network structure matters for contagion
Individual degrees affect who is infected and when
Real life networks portray some universal properties
Heavy-tailed degree distribution
Small diameter compared to the size of the population
High clustering compared to the average degree
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Questions?